Multi-mode cavities for high-efficiency nonlinear wavelength conversion formed with overlap optimization

ABSTRACT

A dual frequency optical resonator configured for optical coupling to light having a first frequency ω 1 . The dual frequency optical resonator includes a plurality of alternating layer pairs stacked in a post configuration, each layer pair having a first layer formed of a first material and a second layer formed of a second material, the first material and second materials being different materials. The first layer has a first thickness and the second layer has a second thickness, the thicknesses of the first and second layer being selected to create optical resonances at the first frequency ω 1  and a second frequency ω 2  which is a harmonic of ω 1  and the thicknesses of the first and second layer also being selected to enhance nonlinear coupling between the first frequency ω 1  and a second frequency ω 2.

CROSS-REFERENCE TO PRIOR FILED APPLICATIONS

This application claims priority to U.S. provisional application62/300,516, filed Feb. 26, 2016, which is incorporated herein in itsentirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under Grant No.DGE1144152 awarded by the National Science Foundation. The governmenthas certain rights in the invention

FIELD OF THE INVENTION

The present disclosure generally relates to the field ofmicro/nano-scale devices and in more particular, micro/nano-scaledevices which can be used for high-efficiency optical nonlinearwavelength conversion in multi-mode cavities.

BACKGROUND

High-efficiency coherent wavelength conversion is important to variousareas of science and technology such as LEDs and lasers, spectroscopy,microscopy and quantum information processing. Current technologiesemploy wavelength converters with bulky nonlinear crystals (e.g. LiNbO3)to convert light at readily available wavelengths to desiredwavelengths. Developing ultra-compact converters with dimensions on thescale of the wavelength of light itself (sub-micron to a few microns)has been hampered by the lack of viable design techniques that canidentify optimal geometries for such devices. This technique canautomatically define optimal geometries that meet the stringentrequirements of high-efficiency wavelength conversion in ultra-compactdevices. A novel micro-post cavity with alternating material layersdeployed in an unusual aperiodic sequence is used to support modes withthe requisite frequencies, large lifetimes, small modal volumes, andextremely large overlaps. This leads to orders of magnitude enhancementsin second harmonic generation. An important advantage of this technologyis faster operational speeds (or more operational bandwidths) overcurrent devices for comparable or even better performance.

SUMMARY OF THE INVENTION

A dual frequency optical resonator configured for optical coupling tolight having a first frequency ω1 is disclosed. The dual frequencyoptical resonator includes a plurality of alternating layer pairsstacked in a post configuration, each layer pair having a first layerformed of a first material and a second layer formed of a secondmaterial, the first material and second materials being differentmaterials. The first layer has a first thickness and the second layerhas a second thickness, the thicknesses of the first and second layerbeing selected to create optical resonances at the first frequency ω1and a second frequency ω2 which is a harmonic of ω1 and the thicknessesof the first and second layer also being selected to enhance nonlinearcoupling between the first frequency ω1 and a second frequency ω2.

The second frequency ω2 may be a harmonic such as a second or thirdharmonic of the first frequency ω1. The thicknesses of the first andsecond layer may be selected to maximize the nonlinear coupling betweenthe first frequency ω1 and a second frequency ω2. The first material maybe AlGaAs and the second material may be Al2O3. The first and secondlayer may be formed in a deposition process.

Another dual frequency optical resonator configured for optical couplingto light having a first frequency ω1 is also disclosed. The dualfrequency optical resonator includes a plurality of alternating layerspairs configured in a grating configuration, each layer pair having afirst layer formed of a first material and a second layer formed of asecond material, the first material and second materials being differentmaterials. The first layer has a first thickness and the second layerhas a second thickness, the thicknesses of the first and second layerbeing selected to create optical resonances at the first frequency ω1and a second frequency ω2 which is a harmonic of ω1 and the thicknessesof the first and second layer also being selected to enhance nonlinearcoupling between the first frequency ω1 and a second frequency ω2.

The second frequency ω2 may be a harmonic such as a second or thirdharmonic of the first frequency ω1. The thicknesses of the first andsecond layer may be selected to maximize the nonlinear coupling betweenthe first frequency ω1 and a second frequency ω2. The first material maybe AlGaAs and the second material may be Al2O3. The first material maybe GaAs and the second material is SiO2. The first material may be LNand the second material may be air. The first and second layer may beformed in an etching process.

Another dual frequency optical resonator configured for optical couplingto light having a first frequency ω1 is also disclosed. The dualfrequency optical resonator includes a plurality pixels configured in anX-Y plane, each pixel being formed of either a first material or asecond material, the first material and second materials being differentmaterials. The material for each pixel is selected such that theplurality of pixels create optical resonances at the first frequency caland a second frequency ω2 which is a harmonic of ω1 and the material foreach pixel is also selected such that the plurality of pixels enhancenonlinear coupling between the first frequency ω1 and a second frequencyω2.

The second frequency ω2 may be a harmonic such as a second or thirdharmonic of the first frequency cal. The material for each pixel may beselected such that the plurality of pixels maximize the nonlinearcoupling between the first frequency ω1 and a second frequency ω2. Thefirst material may be GaAs and the second material may be air. The firstmaterial may be LN and the second material may be air.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1A is a block diagram of a dual frequency rectangular micropostcavity;

FIGS. 1B and 1C are graphs that plot the y-components of the electricfields in the xz-plane of the structure of FIG. 1A;

FIG. 1D is a dual frequency GaAs grating structure;

FIG. 1E is a graph of the cross-sectional dielectric profile of thestructure of FIG. 1D;

FIGS. 1F-1G are graphs that plot the y-components of the electric fieldsin the xz-plane of the structure of FIG. 1D;

FIG. 1H is a dual frequency lithium-niobate (LN) grating structure inair;

FIG. 1I is a graph of the cross-sectional dielectric profile of thestructure of FIG. 1H;

FIGS. 1J-1K are graphs that plot the y-components of the electric fieldsin the xz-plane of the structure of FIG. 1H;

FIG. 2 is a graph showing a trend among various geometries towardsincreasing β and decreasing Q^(rad) as device sizes decrease;

FIGS. 3A-3C show a block diagram of the work flow of the design process;

FIGS. 4A-4B are a schematic illustration of topology-optimizedmultitrack ring resonators;

FIGS. 5A-5D show the statistical distribution of lifetimes Q1,2,frequency mismatch Δω=|ω1−ω2/2|, and nonlinear coupling β, correspondingto the multi-track ring of FIG. 4 associated with the azimuthal modepair (6, 12);

FIG. 6 presents a proof-of-concept 2D design that satisfies all of theserequirements;

FIG. 7A is a diagram of a large-area (non-cavity based) device;

FIG. 7B-7C are graphs that plot the FF mode and SH modes of thestructure of FIG. 7A; and

FIG. 7D is a graph that plots Re[E_(z)] of the structure of FIG. 7A.

DETAILED DESCRIPTION

Nonlinear optical processes mediated by second-order (χ⁽²⁾)nonlinearities play a crucial role in many photonic applications,including ultra-short pulse shaping, spectroscopy, generation of novelfrequencies and states of light and quantum information processing.Because nonlinearities are generally weak in bulk media, a well-knownapproach for lowering the power requirements of devices is to enhancenonlinear interactions by employing optical resonators that confinelight for long times (dimensionless lifetimes Q) in small volumes V.Microcavity resonators designed for on-chip, infrared applications offersome of the smallest confinement factors available, but theirimplementation in practical devices has been largely hampered by thedifficult task of identifying wavelength-scale (V˜λ³) structuressupporting long-lived, resonant modes at widely separated wavelengthsand satisfying rigid frequency-matching and mode-overlap constraints.

This disclosure is directed to scalable topology optimization ofmicrocavities, where every pixel of the geometry is a degree of freedomand to the problem of designing wavelength-scale photonic structures forsecond harmonic generation (SHG). This approach is applied to obtainnovel micropost, and grating microcavity designs supporting stronglycoupled fundamental and harmonic modes at infrared and visiblewavelengths with relatively large lifetimes Q₁, Q₂>10⁴. In contrast torecently proposed designs based on known, linear cavity structureshand-tailored to maximize the Purcell factors or mode volumes ofindividual resonances, e.g. ring resonators and nanobeam cavities, thedisclosed designs ensure frequency matching and small confinementfactors while also simultaneously maximizing the SHG enhancement factorQ² Q₂|β|² to yield orders of magnitude improvements in the nonlinearcoupling β and determined by a special overlap integral between themodes. These particular optimizations of multilayer stacks illustratethe benefits in an approachable and experimentally feasible setting,laying the framework for future topology optimization of 2D/3D slabstructures that are sure to yield even further improvements.

TABLE I SHG figures of merit for topology-optimized micropost andgrating cavities of different material systems. Structure h_(x) × h_(y)× h_(z) (λ₁ ³) λ (μm) (Q₁, Q₂) (Q₁ ^(rad), Q₂ ^(rad)) β FOM₁ FOM₂ (1)AlGaAs/Al₂O₃ micropost 8.4 × 3.5 × 0.84 1.5-0.75 (5000, 1000) 1.4 × 10⁵,1.3 × 10⁵) 0.018 7.5 × 10⁶ 8.3 × 10¹¹ (2) GaAs gratings in SiO₂ 5.4 ×3.5 × 0.60 1.8-0.9 (5000, 1000) (5.2 × 10⁴, 7100) 0.020   7 × 10⁶ 7.5 ×10⁹ (3) LN gratings in air 5.4 × 3.5 × 0.80 0.8-0.4 (5000, 1000) (6700,2400) 0.030 8.4 × 10⁵ 9.7 × 10⁷

Most experimental demonstrations of SHG in chip-based photonic systemsoperate in the so-called small-signal regime of weak nonlinearities,where the lack of pump depletion leads to the well-known quadraticscaling of harmonic output power with incident power. In situationsinvolving all-resonant conversion, where confinement and longinteraction times lead to strong nonlinearities and non-negligible downconversion, the maximum achievable conversion efficiency

$\begin{matrix}{\left( {\eta \equiv \frac{P_{2}^{out}}{P_{1}^{in}}} \right),{\eta^{\max} = {\left( {1 - \frac{Q_{1}}{Q_{1}^{rad}}} \right)\left( {1 - \frac{Q_{2}}{Q_{2}^{rad}}} \right)}}} & (1)\end{matrix}$

occurs at a critical input power,

$\begin{matrix}{{P_{1}^{crit} = {\frac{2\omega_{1}\varepsilon_{0}\lambda_{1}^{3}}{\left( \chi_{eff}^{(2)} \right)^{2}{\overset{\_}{\beta}}^{2}Q_{1}^{2}Q_{2}}\left( {1 - \frac{Q_{1}}{Q_{1}^{rad}}} \right)^{- 1}}},} & (2)\end{matrix}$

where X_(eff) ⁽²⁾ is the effective nonlinear susceptibility of themedium [SM],

$Q = \left( {\frac{1}{Q^{rad}} + \frac{1}{Q^{c}}} \right)^{- 1}$

is the dimensionless quality factor (ignoring material absorption)incorporating radiative decay

$\frac{1}{Q^{rad}}$

and coupling to an input/output channel

$\frac{1}{Q^{c}}.$

The dimensionless coupling coefficient β is given by a complicated,spatial overlap-integral involving the fundamental and harmonic modes[SM],

$\begin{matrix}{{\overset{\_}{\beta} = {\frac{\int{d_{r}{\overset{\_}{\varepsilon}(r)}E_{2}^{*}E_{1}^{2}}}{\left( {\int{{dr}\; \varepsilon_{1}{E_{1}}^{2}}} \right)\left( \sqrt{\int{{dr}\; \varepsilon_{2}{E_{2}}^{2}}} \right)}\sqrt{\lambda_{1}^{3}}}},} & (3)\end{matrix}$

Where ∈ (r)=1 inside the nonlinear medium and zero elsewhere. Based onthe above expressions one can define the following dimensionless figuresof merit

$\begin{matrix}{{{FOM}_{1} = {Q_{1}^{2}Q_{2}{\overset{\_}{\beta}}^{2}\left( {1 - \frac{Q_{1}}{Q_{1}^{rad}}} \right)^{2}\left( {1 - \frac{Q_{2}}{Q_{2}^{rad}}} \right)}},} & (4) \\{{FOM}_{2} = {\left( Q_{1}^{rad} \right)^{2}Q_{2}^{rad}{{\overset{\_}{\beta}}^{2}.}}} & (5)\end{matrix}$

where FOM₁ represents the efficiency per power, often quoted in theso-called undepleted regime of low-power conversion, and FOM₂ representslimits to power enhancement. Note that for a given radiative loss rate,FOM₁ is maximized when the modes are critically coupled,

${Q = \frac{Q^{rad}}{2}},$

with the absolute maximum occurring in the absence of radiative losses,Q^(rad)→∞, or equivalently, when FOM₂ is maximized. From either FOM, itis clear that apart from frequency matching and lifetime engineering,the design of optimal SHG cavities rests on achieving a large nonlinearcoupling β (non-linear overlap).

Optimal Designs.—

Table I characterizes the FOMs of some of our newly discoveredmicrocavity designs, involving simple micropost and gratings structuresof various χ⁽²⁾ materials, including GaAs, AlGaAs and LiNbO₃. Thelow-index material layers of the microposts consist of alumina (Al₂O₃),while gratings are embedded in either silica or air (see supplement fordetailed specifications). Note that in addition to their performancecharacteristics, these structures are also significantly different fromthose obtained by conventional methods in that traditional designs ofteninvolve rings, periodic structures or tapered defects, which tend toignore or sacrifice β in favor of increased lifetimes and for which itis also difficult to obtain widely separated modes.

FIG. 1A is a block diagram of an optimized structure—a doubly-resonantrectangular micropost cavity (micropost resonator) 20 including aplurality of alternating layer pairs 22 stacked in a post configuration.The micropost resonator 20 in this example has only a single dimensionof variation, the thickness of each layer. Each layer pair has a firstlayer 24 formed of a first material and a second layer 26 formed of asecond material. The first material and second materials are differentmaterials and in this example the micropost resonator 20 usesalternating AlGaAs/Al₂O₃ layers along with spatial profiles of thefundamental and harmonic modes. It differs from conventional micropostsin that it does not use periodic bi-layers (e.g., based on a hand gapapproach as the case would be in a DBR device) yet it supports twolocalized modes at precisely λ₁=1.5 μm and λ₂=λ₁/2. In addition tohaving large Q^(rad)≳10⁵ and small V˜(λ₁/n)³, the structure exhibits anultra-large nonlinear coupling β≈0.018 that is almost an order ofmagnitude larger than the best overlap found in the literature (seee.g., FIG. 2). From an experimental point of view, the micropost, systemis of particular interest because it can be realized by a combination ofexisting fabrication techniques such as molecular beam epitaxy, atomiclayer deposition, selective oxidation and electron-beam lithography.Additionally, the micropost cavity can be naturally integrated withquantum dots and quantum wells for cavity QED applications. Similar toother wavelength-scale structures, the operational bandwidths of thesestructures are limited by radiative losses in the lateral direction, buttheir ultra-large overlap factors more than compensate for the increasedbandwidth, which ultimately may prove beneficial in experiments subjectto fabrication imperfections and for large-bandwidth applications.

It should be understood that other structures having a single dimensionof freedom or multiple dimensions of freedom may be used withoutdeparting from the scope of this disclosure. For example, FIG. 1E is agraph of the cross-sectional dielectric profile of the structure of FIG.1D. FIGS. 1F-1G are graphs that plot the y-components of the electricfields in the xz-plane of the structure of FIG. 1DC. FIG. 1I-1 is a dualfrequency LN grating structure in air. FIG. 1I is a graph of thecross-sectional dielectric profile of the structure of FIG. 111. FIGS.1J-1K are graphs that plot the y-components of the electric fields inthe xz-plane of the structure of FIG. 111.

To understand the mechanism of improvement in β, it is instructive toconsider the spatial profiles of interacting modes. FIGS. 1B and 1C plotthe y-components of the electric fields in the xz-plane against thebackground structure. Since β is a net total of positive and negativecontributions coming from the local overlap factor E₁ ²E₂ in thepresence of nonlinearity, not all local contributions are useful for SHGconversion. Most notably, one observes that the positions of negativeanti-nodes of E₂ (light red regions) coincide with either the nodes ofE₁ or alumina layers where x⁽²⁾=0), minimizing negative contributions tothe integrated overlap. In other words, improvements in β do not arisepurely due to tight modal confinement but also from the constructiveoverlap of the modes enabled by the strategic positioning of fieldextrema along the structure.

Based on the tabulated FOMs (Table I), the efficiencies and powerrequirements of realistic devices can be directly calculated. Forexample, assuming x_(eff) ² (AlGaAs)˜100 pm/V, the AlGaAs/Al₂O₃micropost cavity (FIGS. 1A and 1B) yields an efficiency of

$\frac{P_{2,{out}}}{P_{1}^{2}} = {2.7 \times {10^{4}/W}}$

in the undepleted regime when the modes are critically coupled,

$Q = {\frac{Q^{rad}}{2}.}$

For larger operational bandwidths, e.g. Q₁=5000 and Q₂=1000, we findthat

$\frac{P_{2,{out}}}{P_{1}^{2}} = {16/{W.}}$

When the system is in the depleted regime and critically coupled, wefind that a maximum efficiency of 25% can be achieved at P₁ ^(crit)≈0.15mW whereas assuming smaller Q₁=5000 and Q₂=1000, a maximum efficiency of96% can be achieved at P₁ ^(crit)≈0.96 W.

Comparison against previous designs.—Table II summarizes variousperformance characteristics, including the aforementioned FOM, for ahandful of previously studied geometries with length-scales spanningfrom mm to a few wavelengths (microns). FIG. 2 demonstrates a trendamong these geometries towards increasing β and decreasing Q^(rad) asdevice sizes decrease. Maximizing β in millimeter-to-centimeter scalebulky media translates to the well-known problem of phase-matching themomenta or propagation constants of the modes. In this category,traditional WGMRs offer a viable platform for achieving high-efficiencyconversion; however, their ultra-large lifetimes (critically dependentupon material-specific polishing techniques), large sizes (millimeterlength-scales), and extremely weak nonlinear coupling (large modevolumes) render them far-from optimal chip-scale devices. Althoughminiature WGMRs such as microdisk and microring resonators showincreased promise due to their smaller mode volumes, improvements in βare still hardly sufficient for achieving high efficiencies at lowpowers. Ultra-compact nanophotonic resonators such as the recentlyproposed nanorings, 2D pho-tonic crystal defects, and nanobeam cavities,possess even smaller mode volumes but prove challenging for design dueto the difficulty of finding well-confined modes at both the fundamentaland second harmonic frequencies. Even when two such resonances can befound by fine-tuning a limited set of geometric parameters, thefrequency-matching constraint invariably leads to sub-optimal spatialoverlaps which severely limits the maximal achievable β.

Comparing Tables I and II, one observes that for a comparable Q, thetopology-optimized structures perform significantly better in both FOM₁and FOM2 than any conventional geometry, with the exception of the LNgratings, whose low Q^(rad) lead to slightly lower FOM2. Generally, theoptimized microposts and gratings perform better by virtue of a largeand robust β which, notably, is significantly larger than that ofexisting designs. Here, we have not included in our comparison thosestructures which achieve non-negligible SHG by special poling techniquesand/or quasi-phase matching methods, though their performance is stillsub-optimal compared to the topology-optimized designs. Such methods arehighly material-dependent and are thus not readily applicable to othermaterial platforms; instead, ours is a purely geometrical topologyoptimization technique applicable to any material system.

Optimization Formulation:

Optimization techniques have been regularly employed by the photonicdevice community, primarily for fine-tuning the characteristics of apre-determined geometry; the majority of these techniques involveprobabilistic Monte-Carlo algorithms such as particle swarms, simulatedannealing and genetic algorithms. While some of these gradient-freemethods have been used to uncover a few unexpected results out of alimited number of degrees of freedom (DOF), gradient-based topologyoptimization methods efficiently handle a far larger design space,typically considering every pixel or voxel as a DOF in an extensive 2Dor 3D computational domain, giving rise to novel topologies andgeometries that might have been difficult to conceive from conventionalintuition alone. The early applications of topology optimization wereprimarily focused on mechanical problems and only recently have theybeen expanded to consider photonic systems, though largely limited tolinear device designs.

TABLE II Structure λ (μm) (Q₁, Q₂) (Q₁ ^(rad), Q₂ ^(rad)) β FOM₁ FOM₂ LNWGM resonator 1.064-0.532  (3.4 × 10⁷, —) (6.8 × 10⁷, —) —  ~10¹⁰ — AINmicroring 1.55-0.775 (~10⁴, ~5000) — — 2.6 × 10⁵ — GaP PhC slab*1.485-0.742  (≈6000, —) — —  ≈2 × 10⁵ — GaAs PhC nanobeam  1.7-0.91^(†)(5000, 1000) (>10⁶, 4000) 0.00021 820 1.8 × 10⁸ 1.8-0.91 (5000, 1000) (6× 10⁴, 4000) 0.00012 227 2.1 × 10⁵ AlGaAs nanoring 1.55-0.775 (5000,1000) (10⁴, >10⁶) 0.004   10⁵ 1.6 × 10⁹

Table II includes SHG figures of merit, including the frequencies λ,overall and radiative quality factors Q, Q^(rad) and nonlinear couplingβ of the fundamental and harmonic modes, of representative geometries.Also shown are the FOM₁ and FOM₂ figures of merit described in equations(4) and (5). * SHG occurs between a localized defect mode (at thefundamental frequency) and an extended index guided mode of the PhC.†Resonant frequencies are mismatched.

A high level example of a suitable computation system generally proceedsas follows:

1(a) define a grid of degrees of freedom (DOF). 1(b) assign permittivity(material property) to each DOF. 2(a) place a dipole current source J₁at ω₁ in the domain and compute a relative electric field E₁ by solvingMaxwell's equations. 2(b) compute the derivative of E₁ with respect toeach DOF. 3(a) using E₁ at ω1, compute the work done by the electricfield on the current source (P=E₁·J₁). 3b) compute the field E₂ due tocurrent source J2 at ω₂ (e.g., 2 ω₁ for the 2^(nd) harmonic) by solvingMaxwell's equations. 3(c) compute the work done by the electric field onthe current source (P=E₂·J₂). 4 maximize 3(c) and 3(a). In this exampleβ is proportional to 3(c) and 3(a) and 3(c) also ensure that there are 2resonances at ω₁ and ω₂.

In what follows, we describe a system for gradient-based topologyoptimization of nonlinear wavelength-scale frequency converters.Previous approaches exploited the equivalency between LDOS and the powerradiated by a point dipole in order to reduce Purcell-factormaximization problems to a series of small scattering calculations.Defining the objective max _(∈) ƒ(∈(r); ω)=−Re[∫dr J*·E] it follows thatE can be found by solving the frequency domain Maxwell's equationsME=iωJ, where M is the Maxwell operator [SM] and J=δ(r−r₀)êj. Themaximization is then performed over a finely discretized space definedby the normalized dielectric function {∈ _(α)=∈(r_(α)), α

(iΔx, jΔy, kΔz)}. An important realization is that instead of maximizingthe LDOS at a single discrete frequency ω, a better-posed problem isthat of maximizing the frequency-averaged ƒ in the vicinity of ω,denoted by (ƒ)=∫dω′W(ω′;ω,Γ)ƒ(ω′), where W is a weight function definedover some specified bandwidth Γ. Using contour integration techniques,the frequency integral can be conveniently replaced by a singleevaluation of ƒ at a complex frequency ω+iΓ. For a fixed Γ, thefrequency average effectively shifts the algorithm in favor ofminimizing V over maximizing Q; the latter can be enhanced over thecourse of the optimization by gradually winding down the averagingbandwidth Γ. A major merit of the frequency-averaged LDOS formulation isthat it features a mathematically well-posed objective as opposed to adirect maximization of the cavity Purcell factor Q, allowing for rapidconvergence of the optimization algorithm into an extremal solution.

An extension of the optimization problem from single to multimodecavities maximizes the minimum of a collection of LDOS at differentfrequencies. Applying such an approach to the problem of SHG, theoptimization objective becomes: max _(∈) _(α) min [LDOS(ω₁), LDOS(2ω₂)which would require solving two separate scattering problems, M₁E₁=J₁and M₂E₂=J₂, for the two distinct point sources J₁, J₂ at ω₁ and ω₂=2ω₁respectively. However, as discussed before, rather than maximizing thePurcell factor at individual resonances, the key to realizing optimalSHG is to maximize the overlap integral β between E₁ and E₂. Here, wedisclose an elegant way to incorporate β by coupling the two scatteringproblems. In particular, we consider not a point dipole but an extendedsource J₂˜E₁ ² at ω₂ and optimize a single combined radiated powerf=Re[∫dr J₂*·E₂] instead of two otherwise unrelated LDOS. The advantageof this approach is that f yields precisely the β parameter along withany resonant enhancement factors (˜Q/V) in E₁ and E₂. Intuitively, J₂can be thought of as a nonlinear polarization current induced by E₁ inthe presence of the second order susceptibility tensor X⁽²⁾, and inparticular is given by J_(2i)=∈(r)Σ_(jk)x_(ijk) ⁽²⁾ E_(1j)E_(1k) wherethe indices i, j, k run over the Cartesian coordinates. In general,x_(ijk) ⁽²⁾ mixes polarizations and hence ƒ is a sum of differentcontributions from various polarization-combinations. In what followsand for simplicity, we focus on the simplest case in which E₁ and E₂have the same polarization, corresponding to a diagonal X⁽²⁾ tensordetermined by a scalar x_(eff) ⁽²⁾. Such an arrangement can be obtainedfor example by proper alignment of the crystal orientation axes [SM].With this simplification, the generalization of the lineartopology-optimization problem to the case of SHG becomes:

$\begin{matrix}{{{\max\limits_{{\overset{\_}{\varepsilon}}_{\alpha}}{\langle{f\left( {{\overset{\_}{\varepsilon}}_{\alpha};\omega_{1}} \right)}\rangle}} = {- {{Re}\left\lbrack {\langle{\int{{J_{2}^{*} \cdot E_{2}}{dr}}}\rangle} \right\rbrack}}},{{\mathcal{M}_{1}E_{1}} = {i\; \omega_{1}J_{1}}},{{\mathcal{M}_{2}E_{2}} = {i\; \omega_{2}J_{2}}},{\omega_{2} = {2\omega_{1}}}} & (6)\end{matrix}$

where

J₁ = δ(r_(α) − r₀)ê_(j), j ∈ {x, y, z}${J_{2} = {{\overset{\_}{\varepsilon}\left( r_{\alpha} \right)}E_{1\; j}^{2}{\hat{e}}_{j}}},{\mathcal{M}_{l} = {\nabla{\times \frac{1}{\mu}{\nabla{\times {- {\varepsilon_{l}\left( r_{\alpha} \right)}}\omega_{l}^{2}}}}}},{l = 1},2$${{\varepsilon_{l}\left( r_{\alpha} \right)} = {\varepsilon_{m} + {{\overset{\_}{\varepsilon}}_{\alpha}\left( {\varepsilon_{dl} - \varepsilon_{m}} \right)}}},{{\overset{\_}{\varepsilon}}_{\alpha} \in \left\lbrack {0,1} \right\rbrack},$

and where ∈_(d) denotes the dielectric contrast of the nonlinear mediumand ∈_(m) is that of the surrounding linear medium. Note that ∈ _(α) isallowed to vary continuously between 0 and 1 whereas the intermediatevalues can be penalized by so-called threshold projection filters. Thescattering framework makes it straightforward to calculate thederivatives of ƒ (and possible functional constraints) with respectiveto ∈ _(a) via the adjoint variable method. The optimization problem canthen be solved by any of the many powerful algorithms for convex,conservative, separable approximations, such as the well-known method ofmoving asymptotes.

FIG. 3 is a block diagram of the work flow of the design process. Thedegrees of freedom in our problem consist of all the pixels alongx-direction in a 2D computational domain. Starting from the vacuum or auniform slab, the optimization seeks to develop an optimal pattern ofmaterial layers (with a fixed thickness in the z-direction) that cantightly confine light at the desired frequencies while ensuring maximalspatial overlap between the confined modes. The developed 2Dcross-sectional patterns is truncated at a finite width in they-direction to produce a fully three-dimensional micropost or gratingcavity which is then simulated by FDTD methods to extract the resonantfrequencies, quality factors, eigenmodes and corresponding modaloverlaps. Here, it must be emphasized that we merely performedone-dimensional optimization (within a 2D computational problem) becauseof limited computational resources; consequently, our design space isseverely constrained.

For computational convenience, the optimization is carried out using a2D computational cell (in the xz-plane), though the resulting optimizedstructures are given a finite transverse extension h_(y) (along the ydirection) to make realistic 3D devices (see e.g., FIG. 3). Inprinciple, the wider the transverse dimension, the better the cavityquality factors since they are closer to their 2D limit which onlyconsists of radiation loss in the z direction; however, as h_(y)increases, β decreases due to increasing mode volumes. In practice, wechose h_(y) on the order of a few vacuum wavelengths so as not togreatly compromise either Q or β. We then analyze the 3D structures viarigorous FDTD simulations to determine the resonant lifetimes and modaloverlaps. By virtue of our optimization scheme, we invariably find thatfrequency matching is satisfied to within the mode linewidths. We notethat our optimization method seeks to maximize the intrinsic geometricparameters such as Q^(rad) and β of an un-loaded cavity whereas theloaded cavity lifetime Q depends on the choice of coupling mechanism,e.g. free-space, fiber, or waveguide coupling, and is therefore anexternal parameter that can be considered independently of theoptimization. When evaluating the performance characteristics such asFOM₁, we assume total operational lifetimes Q₁=5000, Q₂=1000. In theoptimized structures, it is interesting to note the appearance of deeplysub-wavelength features

${\sim {1 - {5\% \mspace{14mu} {of}\mspace{14mu} \frac{\lambda_{1}}{n}}}},$

creating a kind of metamaterial in the optimization direction; thesearise during the optimization process regardless of starting conditionsdue to the low-dimensionality of the problem. We find that thesefeatures are not easily removable as their absence greatly perturbs thequality factors and frequency matching.

The computational framework discussed above is based on largescaletopology-optimization (TO) techniques that enable automatic discovery ofmultilayer and grating structures exhibiting some of the largest SHGfigures of merit ever predicted. It is also possible to extend the TOformulation to allow the possibility of more sophisticated nonlinearprocesses and apply it to the problem of designing rotationallysymmetric and slab microresonators that exhibit high-efficiency secondharmonic generation (SHG) and sum/difference frequency generation(SFG/DFG). In particular, disclosed herein are multi-track ringresonators and proof-of-principle two-dimensional slab cavitiessupporting multiple, resonant modes (even several octaves apart) thatwould be impossible to design “by hand”. The disclosed designs ensurefrequency matching, long radiative lifetimes, and small(wavelength-scale) modal confinement while also simultaneouslymaximizing the nonlinear modal overlap (or “phase matching”) necessaryfor efficient NFC. For instance, disclosed herein are topology-optimizedconcentric ring cavities exhibiting SHG efficiencies as high as P₂/P₁²=1.3×10²⁵ (x⁽²⁾)²[W⁻¹] even with low operational Q˜10⁴, a performancethat is on a par with recently fabricated 60 μm-diameter, ultrahighQ˜10⁶ AIN microring resonators (P₂/P₁ ²=1.13×10²⁴ (x⁽²⁾)²[W⁻¹]);essentially, our topology-optimized cavities not only possess thesmallest possible modal volumes ˜(λ/n)³, but can also operate over widerbandwidths by virtue of their increased nonlinear modal overlap.

A typical topology optimization problem seeks to maximize or minimize anobjective function ƒ, subject to certain constraints g, over a set offree variables or degrees of freedom (DOF):

max/min ƒ(∈ _(α))  (1)

g(∈ _(α))≦0  (2)

0≦∈ _(α)≦1  (3)

where the DOFs are the normalized dielectric constants.

∈ _(α)∈[0,1] assigned to each pixel or voxel (indexed α) in a specifiedvolume. The subscript α denotes appropriate spatial discretizationr→(i,j,k)_(α)Δ with respect to Cartesian or curvilinear coordinates.Depending on the choice of background (bg) and structural materials, ∈_(α) is mapped onto position-dependent dielectric constant via∈_(α)=(∈−∈_(bg))∈ _(α)+∈_(bg). The binarity of the optimized structureis enforced by penalizing the intermediate values ∈∈ (0,1) or utilizinga variety of filter and regularization methods. Starting from a randominitial guess or completely uniform space, the technique discoverscomplex structures automatically with the aid of powerful gradient-basedalgorithms such as the method of moving asymptotes (MMA). For anelectromagnetic problem, ƒ and g are typically functions of the electricE or magnetic H fields integrated over some region, which are in turnsolutions of Maxwell's equations under some incident current or field.In what follows, we exploit direct solution of Maxwell's equations,

$\begin{matrix}{{{{\nabla{\times \frac{1}{\mu}{\nabla{\times E}}}} - {{\varepsilon (r)}\omega^{2}E}} = {i\; \omega \; J}},} & (4)\end{matrix}$

describing the steady-state E(r; ω) in response to incident currentsJ(r, θ) at frequency ω. While solution of (4) is straightforward andcommonplace, an important aspect to making optimization problemstractable is to obtain a fast-converging and computationally efficientadjoint formulation of the problem. Within the scope of TO, thisrequires efficient calculations of the derivatives

$\frac{\partial f}{\partial{\overset{\_}{\varepsilon}}_{\alpha}},\frac{\partial g}{\partial{\overset{\_}{\varepsilon}}_{\alpha}}$

at every pixel α, which we perform by exploiting the adjoint-variablemethod (AVM).

Any NFC process can be viewed as a frequency mixing scheme in which twoor more constituent photons at a set of frequencies {ω_(n)} interact toproduce an output photon at frequency Ω=Σ_(n)c_(n)w_(n), where {c_(n)}can be either negative or positive, depending on whether thecorresponding photons are created or destroyed in the process. Given anappropriate nonlinear tensor component X_(ijk) . . . , with i, j, k, . .. ∈{x, y, z}, mediating an interaction between the polarizationcomponents E_(i)(Ω) and E_(1j), E_(2k), . . . , we begin with acollection of point dipole currents, each at the constituent frequencyω_(n), n∈{1, 2, . . . } and positioned at the center of thecomputational cell r′, such that J_(n)=ê_(nv)δ (r−r′), where ê_(nv)∈{ê_(1j), ê_(2k), . . . } is a polarization vector chosen so as toexcite the desired electric field polarization components (v) of thecorresponding mode. Given the choice of incident currents Jn, we solveMaxwell's equations to obtain the corresponding constituentelectric-field response E_(n), from which one can construct a nonlinearpolarization current J(Ω)=∈(r)Π_(n)E_(nv) ^(|cn|(*))ê_(i) whereE_(nv)=E_(n)·ê_(nv) and J(Ω) can be generally polarized (ê₁) in a(chosen) direction that differs from the constituent polarizationsê_(nv). Here, (*) denotes complex conjugation for negative c_(n) and noconjugation otherwise. Finally, maximizing the radiated power, −Re[∫RJ(Ω)*·E(Ω)dr], due to J(Ω), one is immediately led to the followingnonlinear topology optimization (NLTO) problem:

$\begin{matrix}{{{\max\limits_{\overset{\_}{\varepsilon}}{f\left( {\overset{\_}{\varepsilon};\omega_{n}} \right)}} = {- {{Re}\left\lbrack {\int{{{J(\Omega)}^{*} \cdot {E(\Omega)}}{dr}}} \right\rbrack}}},{{{\mathcal{M}\left( {\overset{\_}{\varepsilon},\omega_{n}} \right)}E_{n}} = {i\; \omega_{n}J_{n}}},{J_{n} = {{\hat{e}}_{nv}{\delta \left( {r - r^{\prime}} \right)}}},{{{\mathcal{M}\left( {\overset{\_}{\varepsilon},\Omega} \right)}{E(\Omega)}} = {i\; \Omega \; {J(\Omega)}}},{{J(\Omega)} = {\overset{\_}{\varepsilon}{\prod\limits_{n}\; {E_{nv}^{{c_{n}}{{(*})}}{\hat{e}}_{i}}}}},{{\mathcal{M}\left( {\overset{\_}{\varepsilon},\omega} \right)} = {\nabla{\times \frac{1}{\mu}{\nabla{\times {- {\varepsilon (r)}}\omega^{2}}}}}},{{\varepsilon (r)} = {\varepsilon_{m} + {\overset{\_}{\varepsilon}\left( {\varepsilon_{d} - \varepsilon_{m}} \right)}}},{\overset{\_}{\varepsilon} \in {\left\lbrack {0,1} \right\rbrack.}}} & (5)\end{matrix}$

Writing down the objective function in terms of the nonlinearpolarization currents, it follows that solution of (5), obtained byemploying any mathematical programming technique that makes use ofgradient information, e.g. the adjoint variable method maximizes thenonlinear coefficient (mode overlap) associated with the aforementionednonlinear optical process.

Multi-track ring resonators—NLTO formulations may be applied to thedesign of rotationally symmetric cavities for SHG. A material platformmay include gallium arsenide (GaAs) thin films cladded in silica. FIG. 4is a schematic illustration of topology-optimized multitrack ringresonators. Also shown as the cross-sectional profiles of several ringresonators, along with those of fundamental and second harmonic modescorresponding to the azimuthal mode pairs (0,0), (6,12) and (10,21),whose increased lifetimes and modal interactions β (Table III) via aχ⁽²⁾ process lead to increased SHG efficiencies. The result of theoptimizations are described in FIG. 4 and Table III, the latter of whichsummarizes the most important parameters, classified according to thechoice of m₁ and m₂, which denote the azimuthal mode numbers offundamental and second harmonic modes, respectively. (Note thatdepending on the polarization of the two modes, different phase matchingconditions must be imposed, e.g., m₂={2m₁, 2m₁±1}, so in ouroptimizations we consider different possible combinations.) Theparameter β is the nonlinear coupling strength between the interactingmodes, which in the case of SHG is given by:

$\begin{matrix}{{\overset{\_}{\beta} = {\frac{\int{d_{r}{\overset{\_}{\varepsilon}(r)}E_{2}^{*}E_{1}^{2}}}{\left( {\int{{dr}\; \varepsilon_{1}{E_{1}}^{2}}} \right)\left( \sqrt{\int{{dr}\; \varepsilon_{2}{E_{2}}^{2}}} \right)}\sqrt{\lambda_{1}^{3}}}},} & (6)\end{matrix}$

TABLE III (m₁, m₂) Polari- zation Q₁ Q₂$\overset{\_}{\beta}\left( \frac{\chi^{(2)}}{\left. {4\sqrt{(}ɛ_{0}\lambda^{3}} \right)} \right)$Thickness (λ₁) (0, 0) (E_(z), E_(z)) 10⁵   3 × 10⁴ 0.041 0.39 (4, 8)(E_(z), E_(z)) 3.1 × 10⁴   3 × 10³ 0.009 0.30  (5, 10) (E_(z), E_(r))  8 × 10³ 3.7 × 10⁴ 0.008 0.18  (6, 12) (E_(z), E_(z)) 9.5 × 10⁴ 2.7 ×10⁴ 0.008 0.18 (10, 20) (E_(z), E_(z)) 10⁶ 1.2 × 10⁴ 0.004 0.22 (10, 21)(E_(z), E_(r)) 1.6 × 10⁶ 7.4 × 10⁴ 0.004 0.24

Table III shows the SHG figures of merit, including azithmuthal numbersm_(1,2), field polarizations, lifetimes Q_(1,2), and nonlinear couplingβ, in units of χ⁽²⁾/4√{square root over ((∈₀)}λ³), corresponding to thefundamental and harmonic modes of various topology-optimized multi-trackring resonators, with cross-sections (illustrated in FIG. 4) determinedby the choice of thicknesses, given in units of λ₁.

TABLE IV     ω₁:ω₂:ω₃     (m₁, m₂, m₃)     Polarization     (Q₁, Q₂, Q₃)$\overset{\_}{\beta}\left( \frac{\chi^{(2)}}{\left. {4\sqrt{(}ɛ_{0}\lambda^{3}} \right)} \right)$  Thickness (λ₁) 1:1.2:2.2 (0, 0, 0) (E_(z), E_(z), E_(z)) (1.8 × 10⁴,1.4 × 10⁴, 7800) 0.031 0.38

Table IV shows Similar figures of merit as in Table III, but formulti-track rings designed to enhance a SFG process involving light atω₁=ω₃−ω₂, ω₂=1.2ω₁, and ω₃=2.2ω₁, with β.

FIGS. 5A-5D show the statistical distribution of lifetimes Q_(1,2),frequency mismatch Δω=|ω₁−ω₂/2|, and nonlinear coupling β, correspondingto the multi-track ring of FIG. 4 associated with the azimuthal modepair (6, 12). The positions of every interface is subject to randomvariations of maximum extent ±36 nm (blue line) or ±54 nm (red line).

In Table. IV, we also consider resonators optimized to enhance a SFGprocess involving three resonant modes, ω₁=ω₃−ω₂, with ω₂=1.2ω₁ andω₃=2.2ω₁. Note that two of these modes are more than an octave apart.

The resulting structures and figures of merit suggest the possibility oforders of magnitude improvements. In particular, we find that thelargest overlap factors β are achieved in the case m₁=m₂=0,corresponding to highly confined modes with peak amplitudes near thecenter of the rings [FIG. 4A], in which case a relatively thickercavity≈0.42λ₁ is required to mitigate out of plane radiation losses.From the optimized Q's and β and assuming λ₁=1.55 μm, we predict a SHGefficiency of P2/P₁ ²=1.3×10²⁵ (χ⁽²⁾)²[W⁻¹]. As expected, both radiativelosses and β decrease with increasing m, as the modes becomeincreasingly delocalized and move away from the center, resulting inlarger mode volumes. Compared to the state-of-the-art microringresonator, whose β˜10⁻³, our structures exhibit consistently largeroverlaps, albeit with decreased radiative lifetimes. The main challengein realizing multi-track designs is that, like photonic crystals andrelated structures that rely on careful interference effects, their Qstend to be more sensitive to perturbations. In the case of centrallyconfined modes with m₁=m₂=0, we observe the appearance of deeplysubwavelength features near the cavity center where the fields aremostly confined. We find that these features are crucial to theintegrity of the modes since they are responsible for the delicateinterference process which cancels outgoing radiation, and thereforetheir absence greatly reduces the quality factors of the modes. Overall,for m₁=m₂=0, we find that for operation with λ₁˜1.55 μm, a fabricationprecision of several nanometers would be necessary to ensure qualityfactors on the order of 10⁵. On the other hand, the optimized designsbecome increasingly robust for larger m₁,m₂>>0 since they have fewersubwavelength features and smaller aspect ratios. FIG. 4 showsdistributions of the most important figures of merit for an ensemble of(m₁=6, m₂=12) cavities subject to random, uniformly-distributedstructural (position and thicknesses) perturbations in the range [−50,50] nm. We find that while the frequency mismatch and overlap factorsare quite robust against variations, the quality factors can decrease to˜10⁴.

Slab Microcavities—

We now consider a different class of structure and NFC process, namelyDFG in slab microcavities. In particular, we consider a χ⁽³⁾ nonlinearprocess satisfying the frequency relation ω_(s)=ω₀−2ω_(b), with ω_(s),ω₀, and ω_(b) denoting the frequencies of signal, emitted, and pumpphotons (see FIG. 6). Such a DFG process has important implications forsingle-photon frequency conversion, e.g. in nitrogen vacancy (NV) colorcenters, where a single NV photon λ₀=637 nm is converted to atelecommunication wavelength λ_(s)=1550 nm by pump light at λ_(b)˜2200nm, requiring resonances that are more than two octave away from oneanother. In other words, the challenge is to design a diamond cavity(n≈2.4) that exhibits three widely separated strongly confined modeswith large nonlinear interactions and lifetimes. FIG. 6 presents aproof-of-concept 2D design that satisfies all of these requirements.Extension to 3D slabs of finite thickness (assuming similar lateralprofiles and vertical confinement˜wavelength), one is led to thepossibility of ultra-large β˜0.2, with

$\begin{matrix}{\overset{\_}{\beta} = {\frac{\int{d_{r}{\overset{\_}{\varepsilon}(r)}E_{0}^{*}E_{b}^{2}E_{8}}}{\sqrt{\int{{dr}\; \varepsilon_{1}{E_{0}}^{2}}}\sqrt{\int{{dr}\; \varepsilon_{s}{E_{s}}^{2}}}\left( {\int{{dr}\; \varepsilon_{b}{E_{b}}^{2}}} \right)}\lambda_{1}^{3}}} & (7)\end{matrix}$

FIG. 6 shows a topology optimized 2D microcavity exhibiting tightlyconfined and widely separated modes (ω_(s), ω_(b), ω₀) that are severaloctaves apart. The modes interact strongly via a χ⁽³⁾ DFG schemedictated by the frequency relation ω_(s)=ω₀−2ω_(b), with ω₀=2.35ω_(s)and ω_(b)=0.68ω_(s), illustrated by the accompanying two-levelschematic.

Note that the lifetimes of these 2D modes are bounded only by the finitesize of our computational cell (and hence are ignored in ourdiscussion), whereas in realistic 3D microcavities, they will be limitedby vertical radiation losses. Despite the two-dimensional aspect of thisslab design, and in contrast to the fully 3D multitrack ring resonatorsabove, these results provide proof of the existence of wavelength-scalephotonic structures that can greatly enhance challenging NFC processes.One example is the NV problem described above, which is particularlychallenging if a monolithic all-diamond approach is desired, in whichcase both single-photon emission and wavelength conversion are to beseamlessly realized in the same diamond cavity. A viable solution thatwas recently proposed is the use of four-wave mixing Bragg scattering(FWM-BS) by way of whispering gallery modes, which are relatively easyto phase-match but suffer from large mode volumes. Furthermore, FWM-BSrequires two pump lasers, at least one of which has a shorter wavelengththan the converted signal photon, which could lead to spontaneousdown-conversion and undesirable noise, degrading quantum fidelity, incontrast to the DFG scheme above, based on a long-wavelength pump.

FIG. 7A is a diagram of a large-area (non-cavity based) device. FIGS.7B-7C are graphs that plot the FF mode and SH modes of the structure ofFIG. 7A. FIG. 7D is a graph that plots Re[Ez] of the structure of FIG.7A. As shown in FIG. 7A, the device is configured with as an XY gridwith a plurality of pixels. Each pixel is configured with either anactive material, e.g., GaAs or a vacuum.

Further disclosure is contained in U.S. provisional application62/300,516, filed Feb. 26, 2016, which is incorporated herein in itsentirety. All references that are cited in U.S. provisional application62/300,516 and the appendix are also incorporated herein in theirentirety. Further disclosure is also provided in Lin et al. “Topologyoptimization of multi-track ring resonators and 2D microcavities fornonlinear frequency conversion”, Physics—Optics, January 2017 which isalso incorporated herein in its entirety. It should be understood thatmany variations are possible based on the disclosure herein. Althoughfeatures and elements are described above in particular combinations,each feature or element can be used alone without the other features andelements or in various combinations with or without other features andelements. The digital processing techniques disclosed herein may bepartially implemented in a computer program, software, or firmwareincorporated in a computer-readable (non-transitory) storage medium forexecution by a general-purpose computer or a processor. Examples ofcomputer-readable storage mediums include a read only memory (ROM), arandom access memory (RAM), a register, cache memory, semiconductormemory devices, magnetic media such as internal hard disks and removabledisks, magneto-optical media, and optical media such as CD-ROM disks,and digital versatile disks (DVDs).

Suitable processors include, by way of example, a general-purposeprocessor, a special purpose processor, a conventional processor, adigital signal processor (DSP), a plurality of microprocessors, one ormore microprocessors in association with a DSP core, a controller, amicrocontroller, Application-Specific Integrated Circuits (ASICs),Field-Programmable Gate Arrays (FPGAs) circuits, any other type ofintegrated circuit (IC), and/or a state machine.

What is claimed is:
 1. A dual frequency optical resonator configured for optical coupling to light having a first frequency ω₁, the dual frequency optical resonator comprising: a plurality of alternating layer pairs stacked in a post configuration, each layer pair having a first layer formed of a first material and a second layer formed of a second material, the first material and second materials being different materials, the first layer having a first thickness and the second layer having a second thickness, the thicknesses of the first and second layer being selected to create optical resonances at the first frequency ω₁ and a second frequency ω₂ which is a harmonic of ω₁ and the thicknesses of the first and second layer also being selected to enhance nonlinear coupling between the first frequency ω₁ and a second frequency ω₂.
 2. The dual frequency optical resonator of claim 1 wherein ω₂ is a second harmonic of ω₁.
 3. The dual frequency optical resonator of claim 1 wherein ω₂ is a third harmonic of ω₁.
 4. The dual frequency optical resonator of claim 1 wherein the thicknesses of the first and second layer are selected to maximize the nonlinear coupling between the first frequency ω₁ and a second frequency ω₂.
 5. The dual frequency optical resonator of claim 1 wherein the first material is AlGaAs and the second material is Al₂O₃.
 6. The dual frequency optical resonator of claim 1 wherein the first and second layer are formed in a deposition process.
 7. A dual frequency optical resonator configured for optical coupling to light having a first frequency ω₁, the dual frequency optical resonator comprising: a plurality of alternating layers pairs configured in a grating configuration, each layer pair having a first layer formed of a first material and a second layer formed of a second material, the first material and second materials being different materials, the first layer having a first thickness and the second layer having a second thickness, the thicknesses of the first and second layer being selected to create optical resonances at the first frequency ω₁ and a second frequency ω₂ which is a harmonic of ω₁ and the thicknesses of the first and second layer also being selected to enhance nonlinear coupling between the first frequency ω₁ and a second frequency ω₂.
 8. The dual frequency optical resonator of claim 7 wherein ω₂ is a second harmonic of ω₁.
 9. The dual frequency optical resonator of claim 7 wherein ω₂ is a third harmonic of ω₁.
 10. The dual frequency optical resonator of claim 7 wherein the thicknesses of the first and second layer are selected to maximize the nonlinear coupling between the first frequency ω₁ and a second frequency ω₂.
 11. The dual frequency optical resonator of claim 7 wherein the first material is AlGaAs and the second material is Al₂O₃.
 12. The dual frequency optical resonator of claim 7 wherein the first material is GaAs and the second material is SiO₂.
 13. The dual frequency optical resonator of claim 7 wherein the first material is LN and the second material is air.
 14. The dual frequency optical resonator of claim 7 wherein the first and second layer are formed in an etching process.
 15. A dual frequency optical resonator configured for optical coupling to light having a first frequency ω₁, the dual frequency optical resonator comprising: a plurality of pixels configured in an X-Y plane, each pixel being formed of either a first material or a second material, the first material and second materials being different materials, the material for each pixel is selected such that the plurality of pixels create optical resonances at the first frequency ω₁ and a second frequency ω₂ which is a harmonic of ω₁ and the material for each pixel is also selected such that the plurality of pixels enhance nonlinear coupling between the first frequency ω₁ and a second frequency ω₂.
 16. The dual frequency optical resonator of claim 15 wherein ω₂ is a second harmonic of ω₁.
 17. The dual frequency optical resonator of claim 15 wherein ω₂ is a third harmonic of ω₁.
 18. The dual frequency optical resonator of claim 15 wherein the material for each pixel is selected such that the plurality of pixels maximize the nonlinear coupling between the first frequency ω₁ and a second frequency ω₂.
 19. The dual frequency optical resonator of claim 15 wherein the first material is GaAs gratings and the second material is air.
 20. The dual frequency optical resonator of claim 15 wherein the first material is LN and the second material is air. 